Theorems · Definition · number theory

Function.Injective.linearOrderedCommMonoidWithZero

{α : Type u_1} →
  [inst : LinearOrderedCommMonoidWithZero α] →
    {β : Type u_3} →
      [inst_1 : Zero β] →
        [inst_2 : Bot β] →
          [inst_3 : One β] →
            [inst_4 : Mul β] →
              [inst_5 : Pow β ℕ] →
                [inst_6 : LE β] →
                  [inst_7 : LT β] →
                    [inst_8 : Max β] →
                      [inst_9 : Min β] →
                        [inst_10 : Ord β] →
                          [DecidableEq β] →
                            [DecidableLE β] →
                              [DecidableLT β] →
                                (f : β → α) →
                                  Function.Injective f →
                                    f 0 = 0 →
                                      f 1 = 1 →
                                        (∀ (x y : β), f (x * y) = f x * f y) →
                                          (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) →
                                            (∀ {x y : β}, f x ≤ f y ↔ x ≤ y) →
                                              (∀ {x y : β}, f x < f y ↔ x < y) →
                                                (∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) →
                                                  (∀ (x y : β), f (x ⊓ y) = min (f x) (f y)) →
                                                    f ⊥ = ⊥ →
                                                      (∀ (x y : β), compare (f x) (f y) = compare x y) →
                                                        LinearOrderedCommMonoidWithZero β

Pullback a LinearOrderedCommMonoidWithZero under an injective map. See note [reducible non-instances].

Defined in

Mathlib.Algebra.Order.GroupWithZero.Canonical

Cited by

0 results in Mathlib

Foundations

Depth 18 from the axioms · uses propext

Assumes

LinearOrderedCommMonoidWithZeroZeroBotOneMulPowLELTMaxMinOrdDecidableEqDecidableLEDecidableLT